Solve sqrt{3^2*{(1-2/3)*[(5/9+1/3)*(3/4-frac{2}{5}-frac{1}{8}):(1-frac{9}{10})]}}

$$\sqrt { 3 ^ { 2 } \times \{ ( 1 - \frac { 2 } { 3 } ) \times [ ( \frac { 5 } { 9 } + \frac { 1 } { 3 } ) \times ( \frac { 3 } { 4 } - \frac { 2 } { 5 } - \frac { 1 } { 8 } ) : ( 1 - \frac { 9 } { 10 } ) ] \} }$$

Evaluate

$$\sqrt{6}\approx 2.449489743$$

Solution Steps

Calculate 3 to the power of 2 and get 9.

$$\sqrt{9\left(1-\frac{2}{3}\right)\times \frac{\left(\frac{5}{9}+\frac{1}{3}\right)\left(\frac{3}{4}-\frac{2}{5}-\frac{1}{8}\right)}{1-\frac{9}{10}}}$$

Convert 1 to fraction \frac{3}{3}.

$$\sqrt{9\left(\frac{3}{3}-\frac{2}{3}\right)\times \frac{\left(\frac{5}{9}+\frac{1}{3}\right)\left(\frac{3}{4}-\frac{2}{5}-\frac{1}{8}\right)}{1-\frac{9}{10}}}$$

Since \frac{3}{3} and \frac{2}{3} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{9\times \frac{3-2}{3}\times \frac{\left(\frac{5}{9}+\frac{1}{3}\right)\left(\frac{3}{4}-\frac{2}{5}-\frac{1}{8}\right)}{1-\frac{9}{10}}}$$

Subtract 2 from 3 to get 1.

$$\sqrt{9\times \frac{1}{3}\times \frac{\left(\frac{5}{9}+\frac{1}{3}\right)\left(\frac{3}{4}-\frac{2}{5}-\frac{1}{8}\right)}{1-\frac{9}{10}}}$$

Multiply 9 and \frac{1}{3} to get \frac{9}{3}.

$$\sqrt{\frac{9}{3}\times \frac{\left(\frac{5}{9}+\frac{1}{3}\right)\left(\frac{3}{4}-\frac{2}{5}-\frac{1}{8}\right)}{1-\frac{9}{10}}}$$

Divide 9 by 3 to get 3.

$$\sqrt{3\times \frac{\left(\frac{5}{9}+\frac{1}{3}\right)\left(\frac{3}{4}-\frac{2}{5}-\frac{1}{8}\right)}{1-\frac{9}{10}}}$$

Least common multiple of 9 and 3 is 9. Convert \frac{5}{9} and \frac{1}{3} to fractions with denominator 9.

$$\sqrt{3\times \frac{\left(\frac{5}{9}+\frac{3}{9}\right)\left(\frac{3}{4}-\frac{2}{5}-\frac{1}{8}\right)}{1-\frac{9}{10}}}$$

Since \frac{5}{9} and \frac{3}{9} have the same denominator, add them by adding their numerators.

$$\sqrt{3\times \frac{\frac{5+3}{9}\left(\frac{3}{4}-\frac{2}{5}-\frac{1}{8}\right)}{1-\frac{9}{10}}}$$

Add 5 and 3 to get 8.

$$\sqrt{3\times \frac{\frac{8}{9}\left(\frac{3}{4}-\frac{2}{5}-\frac{1}{8}\right)}{1-\frac{9}{10}}}$$

Least common multiple of 4 and 5 is 20. Convert \frac{3}{4} and \frac{2}{5} to fractions with denominator 20.

$$\sqrt{3\times \frac{\frac{8}{9}\left(\frac{15}{20}-\frac{8}{20}-\frac{1}{8}\right)}{1-\frac{9}{10}}}$$

Since \frac{15}{20} and \frac{8}{20} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{3\times \frac{\frac{8}{9}\left(\frac{15-8}{20}-\frac{1}{8}\right)}{1-\frac{9}{10}}}$$

Subtract 8 from 15 to get 7.

$$\sqrt{3\times \frac{\frac{8}{9}\left(\frac{7}{20}-\frac{1}{8}\right)}{1-\frac{9}{10}}}$$

Least common multiple of 20 and 8 is 40. Convert \frac{7}{20} and \frac{1}{8} to fractions with denominator 40.

$$\sqrt{3\times \frac{\frac{8}{9}\left(\frac{14}{40}-\frac{5}{40}\right)}{1-\frac{9}{10}}}$$

Since \frac{14}{40} and \frac{5}{40} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{3\times \frac{\frac{8}{9}\times \frac{14-5}{40}}{1-\frac{9}{10}}}$$

Subtract 5 from 14 to get 9.

$$\sqrt{3\times \frac{\frac{8}{9}\times \frac{9}{40}}{1-\frac{9}{10}}}$$

Multiply \frac{8}{9} times \frac{9}{40} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{3\times \frac{\frac{8\times 9}{9\times 40}}{1-\frac{9}{10}}}$$

Cancel out 9 in both numerator and denominator.

$$\sqrt{3\times \frac{\frac{8}{40}}{1-\frac{9}{10}}}$$

Reduce the fraction \frac{8}{40} to lowest terms by extracting and canceling out 8.

$$\sqrt{3\times \frac{\frac{1}{5}}{1-\frac{9}{10}}}$$

Convert 1 to fraction \frac{10}{10}.

$$\sqrt{3\times \frac{\frac{1}{5}}{\frac{10}{10}-\frac{9}{10}}}$$

Since \frac{10}{10} and \frac{9}{10} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{3\times \frac{\frac{1}{5}}{\frac{10-9}{10}}}$$

Subtract 9 from 10 to get 1.

$$\sqrt{3\times \frac{\frac{1}{5}}{\frac{1}{10}}}$$

Divide \frac{1}{5} by \frac{1}{10} by multiplying \frac{1}{5} by the reciprocal of \frac{1}{10}.

$$\sqrt{3\times \frac{1}{5}\times 10}$$

Multiply \frac{1}{5} and 10 to get \frac{10}{5}.

$$\sqrt{3\times \frac{10}{5}}$$

Divide 10 by 5 to get 2.

$$\sqrt{3\times 2}$$

Multiply 3 and 2 to get 6.

$$\sqrt{6}$$